EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5661 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Extension of q-Hermite-Hadamard Inequalities for Strong Convexity Chanokgan Sahatsathatsana1,∗, Pongsakorn Yotkaew1 1 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Abstract. This study leverages q-calculus to establish q-Hermite-Hadamard type inequalities for strongly convex functions, showcasing possible extensions of well-known results in the field. Addi- tionally, it enhances these findings by exploring the strong convexity of the function Φ. Further- more, the q-midpoint and q-trapezoidal inequalities are unified within a comprehensive framework. Ultimately, the results suggest that the newly derived inequalities can be effectively applied in the context of special means. 2020 Mathematics Subject Classifications: 05A30, 26A51, 26D10, 26D15, 26D25, 52A01 Key Words and Phrases: H-H type inequalities, strongly convex functions, quantum calculus 1. Introduction The Hermite-Hadamard inequality (H-H inequality) was proposed and explored by C. Hermite [13] and J. Hadamard [12]. This inequality offers an estimate for the mean value of a convex function and refines the Jensen inequality [9]. In 2018, N. Alp et al. [3] demonstrated a version of the q-H-H inequality for convex functions utilizing left q- integrals. Theorem 1 ([3]). Let Φ : [α,Υ] → R be a convex differentiable function defined on [α,Υ] with 0 < q < 1. The following inequalities then hold: Φ ( qα+Υ [2]q ) ≤ 1 Υ− α ∫ Υ α Φ(ω) αdqω ≤ qΦ(α) + Φ(Υ) [2]q , (1) where [2]q = 1 + q. In 2020, S. Bermudo et al. [5] established the following q-H-H inequality applicable to convex functions. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5661 Email addresses: chanokgan.na@ksu.ac.th (C. Sahatsathatsana), pongyo@kku.ac.th (P. Yotkaew) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 2 of 24 Theorem 2 ([5]). Let Φ : [α,Υ] → R be a convex differentiable function defined on [α,Υ] with 0 < q < 1. The following inequalities are established: Φ ( α+ qΥ [2]q ) ≤ 1 Υ− α ∫ Υ α Φ(ω) Υdqω ≤ Φ(α) + qΦ(Υ) [2]q . (2) By combining the inequalities (1) and (2), the result is presented in [5] as follows: Φ ( α+Υ 2 ) ≤ 1 2(Υ− α) (∫ Υ α Φ(ω) αdqω + ∫ Υ α Φ(ω) Υdqω ) ≤ Φ(α) + Φ(Υ) 2 . (3) In 2023, M.A. Ali et al. [1] and T. Sitthiwirattham et al. [23] formulated the following q-H-H-type inequalities. Theorem 3 ([1, 23]). Let Φ : [α,Υ] → R be a convex function. The following inequalities are valid: Φ ( α+Υ 2 ) ≤ 1 Υ− α (∫ (α+Υ) 2 α Φ(ω) (α+Υ) 2 dqω + ∫ Υ (α+Υ) 2 Φ(ω) (α+Υ) 2 dqω ) ≤ Φ(α) + Φ(Υ) 2 (4) and Φ ( α+Υ 2 ) ≤ 1 Υ− α (∫ (α+Υ) 2 α Φ(ω) αdqω + ∫ Υ (α+Υ) 2 Φ(ω) Υdqω ) ≤ Φ(α) + Φ(Υ) 2 . (5) The H-H inequality has garnered significant attention in the literature, particularly concerning various notions of convexity and extensions and refinements. For further de- tails, interested readers may refer to [6–8] and the references therein. In 1966, B.T. Polyak [19] introduced the principle of strongly convex functions. This idea is significant in mathematical programming and the application of mathematical models. It has been widely utilized in various studies, such as those mentioned in [4, 17, 18, 21, 22]. Definition 1 ([19]). Let Φ : I → R be a function defined on an interval I. We say that Φ is a strongly convex function with modulus φ > 0 if Φ(ϖω + (1−ϖ)ζ) ≤ ϖΦ(ω) + (1−ϖ)Φ(ζ)− φϖ(1−ϖ)(ω − ζ)2 (6) for all ω, ζ ∈ I and ϖ ∈ [0, 1]. Calculus is a vital branch of mathematics focused on the analysis of functions and their continuous changes. In the 17th century, significant advancements in calculus were made by I. Newton and G.W. Leibniz, laying the groundwork for its modern development. Sub- sequently, L. Euler (1707–1783) introduced the concept of quantum calculus (q-calculus), C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 3 of 24 which operates without the traditional notion of limits and forges a link between mathe- matics and physics. In the 20th century, F.H. Jackson [14, 15] further expanded Euler’s ideas by formalizing the principles of q-calculus. Later, in 2000, V.G. Kac and P. Cheung [16] provided a comprehensive overview of the foundational concepts of q-calculus in their publication; additional insights can be found in [10, 11]. In [24, 25], J. Tariboon and S.K. Ntouyas (2013, 2014) introduced the q-calculus for continuous functions defined on finite intervals and examined some of its properties, re- ferred to as qa-calculus. Inspired by the earlier discussion of q-calculus, this study develops q-H-H inequalities for strongly convex functions using the principles of q-calculus. It refines existing findings by utilizing the characterization of strong convexity of Φ. Furthermore, the q-midpoint and q-trapezoidal inequalities are unified into one. The newly established inequalities also demonstrate applications to special means. 2. Preliminaries In this section, we will review the definitions and key properties related to the concept of q-calculus that are pertinent to this study. For the purposes of this paper, we will consider α < Υ and 0 < q < 1 as fixed parameters. [η]q := 1− qη 1− q = 1 + q + q2 + · · ·+ qη−1, η ∈ N. This represents the q-analogue of η; for further information, please refer to [16]. Definition 2 ([16, 24]). Let Φ : [α,Υ] → R be a continuous function. The qα-derivative of Φ at ω ∈ [α,Υ] is, consequently, defined by the following expression αDqΦ(ω) = Φ(ω)− Φ(qω + (1− q)α) (1− q)(ω − α) , ω ̸= α. (7) If ω = α, we define αDqΦ(α) = lim ω→a αDqΦ(ω), whether it exists and is finite. If we take α = 0 in (7), then we have 0DqΦ(ω) = DqΦ(ω), which can be simplified to DqΦ(ω) = Φ(ω)− Φ(qω) (1− q)ω , ω ̸= 0. This represents the q-Jackson derivative. Definition 3 ([5]). Let Φ : [α,Υ] → R be a continuous function. Then, the qΥ-derivative of Φ at ω ∈ [α, b] is defined by ΥDqΦ(ω) = Φ(qω + (1− q)Υ)− Φ(ω) (1− q)(Υ− ω) , ω ̸= Υ. (8) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 4 of 24 If ω = Υ, we define ΥDqΦ(b) = lim ω→Υ ΥDqΦ(ω), whether it exists and is finite. Definition 4 ([20]). Let Φ : [α,Υ] → R be a continuous function and ω ∈ [α,Υ]. The qα-integral of Φ on [α, ω] is defined by∫ ω α Φ(ϖ) αdqϖ = (1− q)(ω − α) ∞∑ n=0 qnΦ (qnω + (1− qn)α) . (9) Note that ∫ ω α Φ(ϖ) αdqϖ = (ω − α) ∫ 1 0 Φ ((1−ϖ)α+ϖω) αdqϖ, and if α = 0, then (9) reduces to∫ ω 0 Φ(ϖ) 0dqϖ = ∫ ω 0 Φ(ϖ)dqϖ = (1− q)ω ∞∑ n=0 qnΦ(qnω). This represents the qα-integral, for further details, refer to [16, 24]. Definition 5 ([5]). Let Φ : [α,Υ] → R be a continuous function and ω ∈ [α,Υ]. The qΥ-integral of Φ on [ω,Υ] is defined by∫ Υ ω Φ(ϖ) Υdqϖ = (1− q)(Υ− ω) ∞∑ n=0 qnΦ (qnω + (1− qn)Υ) . (10) Note that ∫ Υ ω Φ(ϖ) Υdqϖ = (Υ− ω) ∫ 1 0 Φ (ϖΥ+ (1−ϖ)ω) 1dqϖ, and if ω = 0, then (10) reduces to∫ Υ 0 Φ(ϖ) Υdqϖ = (1− q)Υ ∞∑ n=0 qnΦ((1− qn)Υ). This represents the qΥ-integral. 3. Refinements of q-H-H-type inequalities We begin by refining Theorem 3.1 in N. Alp et al. [2] using the characterization of strong convexity. C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 5 of 24 Theorem 4. Let Φ : [α,Υ] → R be a strongly convex function for φ > 0. Then the following inequalities are established: Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ(Υ− α)2 4 ( 4Θ2 [3]q − 4Θ [2]q + 1 ) ≤ 1 2Θ(Υ− α) (∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + ∫ b Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) ≤ Φ(α) + Φ(Υ) 2 − φ(Υ− α)2 ( Θ [2]q − Θ2 [3]q ) ≤ Φ(α) + Φ(Υ) 2 (11) for all Θ ∈ (0, 1]. Proof. It follows from strong convexity of Φ on [α,Υ] that Φ(ϖs+ (1−ϖ)s1) ≤ ϖΦ(s) + (1−ϖ)Φ(s1)− φϖ(1−ϖ)(s− s1) 2, (12) for all s, s1 ∈ [α,Υ] and ϖ ∈ [0, 1]. Substituting ϖ = 1/2 into the inequality (12), then Φ ( s+ s1 2 ) ≤ Φ(s) + Φ(s1) 2 − φ(s− s1) 2 4 . (13) Let Θ ∈ (0, 1] and putting s = ΘϖΥ + (1 − Θϖ)α and s1 = Θϖα + (1 − Θϖ)Υ in the inequality (13), we have Φ ( α+Υ 2 ) ≤ Φ(ΘϖΥ+ (1−Θϖ)α) + Φ(Θϖα+ (1−Θϖ)Υ) 2 − φ(Υ− α)2 (2Θϖ − 1)2 4 . (14) It follows from the definitions 4 and 5, and by applying q-integration to both sides of the inequality (14), we find that Φ ( α+Υ 2 ) ≤ ∫ 1 0 ( Φ(ΘϖΥ+ (1−Θϖ)α) + Φ(Θϖα+ (1−Θϖ)Υ) 2 − φ(Υ− α)2 (2Θϖ − 1)2 4 ) dqϖ = ∫ 1 0 Φ(ΘϖΥ+ (1−Θϖ)α) + Φ(Θϖα+ (1−Θϖ)Υ) 2 dqϖ − ∫ 1 0 φ(Υ− α)2 ( 4Θ2ϖ2 − 4Θϖ + 1 ) 4 dqϖ = (1− q) ∞∑ n=0 qn Φ(ΘqnΥ+ (1−Θqn)α) + Φ(Θqnα+ (1−Θqn)b) 2 C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 6 of 24 − φ(Υ− α)2 4 ( 4Θ2 [3]q − 4Θ [2]q + 1 ) = Θ(Υ− α)(1− q) 2Θ(Υ− α) ( ∞∑ n=0 qnΦ ( (ΘΥ + (1−Θ)α)qn + (1− qn)α ) + ∞∑ n=0 qnΦ ( (Θα+ (1−Θ)Υ)qn + (1− qn)Υ )) − φ(Υ− α)2 4 ( 4Θ2 [3]q − 4Θ [2]q + 1 ) = 1 2Θ(Υ− α) (∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − φ(Υ− α)2 4 ( 4Θ2 [3]q − 4Θ [2]q + 1 ) . (15) Note that 4Θ2/[3]q − 4Θ/[2]q + 1 ≥ 0. This implies that Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ(Υ− α)2 4 ( 4Θ2 [3]q − 4Θ [2]q + 1 ) ≤ 1 2Θ(Υ− α) (∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) . This gives the first and second inequalities of (11). From the inequality (14) and by strong convexity of Φ, we obtain Φ(ΘϖΥ+ (1−Θϖ)α) + Φ(Θϖα+ (1−Θϖ)Υ) 2 − φ(Υ− α)2 (2Θϖ − 1)2 4 ≤ ΘϖΦ(α) + (1−Θϖ)Φ(Υ)− φΘϖ(1−Θϖ)(Υ− α)2 2 + ΘϖΦ(Υ) + (1−Θϖ)Φ(α)− φΘϖ(1−Θϖ)(Υ− α)2 2 − φ(Υ− α)2 (2Θϖ − 1)2 4 = Φ(α) + Φ(Υ) 2 − φ(Υ− α)2 4 . Applying q-integration to both sides of the above inequality yields∫ 1 0 ( Φ(ΘϖΥ+ (1−Θϖ)α) + Φ(Θϖα+ (1−Θϖ)Υ) 2 − φ(Υ− α)2 (2Θϖ − 1)2 4 ) dqϖ ≤ ∫ 1 0 ( Φ(α) + Φ(Υ) 2 − φΘϖ(1−Θϖ)(Υ− α)2 − φ(Υ− α)2 (2Θϖ − 1)2 4 ) dqϖ = Φ(α) + Φ(Υ) 2 − φ(Υ− α)2 4 . (16) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 7 of 24 From the inequalities (15) and (16), we obtain∫ 1 0 ( Φ(ΘϖΥ+ (1−Θϖ)α) + Φ(Θϖα+ (1−Θϖ)b) 2 − φ(Υ− α)2 (2Θϖ − 1)2 4 ) dqϖ = 1 2Θ(Υ− α) (∫ ΘΥ+(1−Θ)α a Φ(ϖ)αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − φ(Υ− α)2 4 ( 4Θ2 [3]q − 4Θ [2]q + 1 ) ≤ Φ(α) + Φ(Υ) 2 − φ(Υ− α)2 4 . Note that Θ/[2]q −Θ2/[3]q ≥ 0. This implies that 1 2Θ(Υ− α) (∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) ≤ Φ(α) + Φ(Υ) 2 − φ(Υ− α)2 ( Θ [2]q − Θ2 [3]q ) ≤ Φ(α) + Φ(Υ) 2 . This finalizes the proof. Remark 1. Setting Θ = 1, in Theorem 4, then we obtain Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ(Υ− α)2 4 ( q3 − 2q2 + 2q − 1 [2]q[3]q ) ≤ 1 2(Υ− α) (∫ Υ α Φ(ϖ)αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) ≤ Φ(α) + Φ(Υ) 2 − φq2(Υ− α)2 [2]q[3]q ≤ Φ(α) + Φ(Υ) 2 , which appeared in [21]. Remark 2. Setting Θ = 1/[2]q, in Theorem 4, leads us to the new q-H-H inequalities: Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ(Υ− α)2 4([2]q)2[3]q ( 4− 4[3]q + ([2]q) 2[3]q ) ≤ [2]q 2(Υ− α) (∫ (qα+Υ) [2]q α Φ(ϖ)αdqϖ + ∫ Υ (α+qΥ) [2]q Φ(ϖ) Υdqϖ ) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 8 of 24 ≤ Φ(α) + Φ(Υ) 2 − φ(Υ− α)2 ([2]q)2[3]q ([3]q − 1) ≤ Φ(α) + Φ(Υ) 2 . Next, we refine Theorem 3.5 from N. Alp et al. [2] by leveraging the properties asso- ciated with strong convexity Theorem 5. Let Φ : [α,Υ] → R be a strongly convex function for φ > 0 and Θ ∈ (0, 1]. Then the following inequalities are established: Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) ≤ 1 + q2 2qΘ[2]q(Υ− α) (∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − 1− q 2q[2]q (Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α)) ≤ Φ(α) + Φ(Υ) 2 − φqΘ(Υ− a)2 [2]q ( 2 [2]q − qΘ [3]q − Θ [3]q ) ≤ Φ(α) + Φ(Υ) 2 . (17) Proof. It follows from strong convexity of Φ on [α,Υ] that Φ ( α+Υ 2 ) = Φ ( α+ qΥ+ qα+Υ 2[2]q ) ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) − φ 4 ( (1− q)(Υ− α) [2]q )2 , which implies that Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) . This proves the first and second inequalities of (17). By strong convexity of Φ, and ϖ ∈ [0, 1] and Θ ∈ (0, 1], we can write Φ ( α+ qΥ [2]q ) = Φ ( (qΘϖΥ+ (1− qΘϖ)α) + q(Θϖα+ (1−Θϖ)Υ) [2]q ) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 9 of 24 = Φ ( (qΘϖΥ+ (1− qΘϖ)α) [2]q + q(Θϖα+ (1−Θϖ)Υ) [2]q ) ≤ Φ(qΘϖΥ+ (1− qΘϖ)α) [2]q + qΦ(Θϖα+ (1−Θϖ)b) [2]q ≤ qΘϖΦ(Υ) + (1− qΘϖ)Φ(α)− φqΘϖ(1− qΘϖ)(Υ− a)2 [2]q + q ( ΘϖΦ(α) + (1−Θϖ)Φ(Υ)− φΘϖ(1−Θϖ)(Υ− α)2 ) [2]q = Φ(α) + qΦ(Υ) [2]q − φqΘ(Υ− α)2 [2]q (2ϖ − qΘϖ2 −Θϖ2), which implies that Φ ( α+ qΥ [2]q ) ≤ Φ(qΘϖΥ+ (1− qΘϖ)α) [2]q + qΦ(Θϖα+ (1−Θϖ)Υ) [2]q ≤ Φ(α) + qΦ(Υ) [2]q − φqΘ(Υ− α)2 [2]q (2ϖ − qΘϖ2 −Θϖ2). By performing q-integration on both sides of the inequality above, we derive Φ ( α+ qΥ [2]q ) ≤ ∫ 1 0 ( Φ(qΘϖΥ+ (1− qΘϖ)α) [2]q + qΦ(Θϖα+ (1−Θϖ)b) [2]q ) dqϖ (18) ≤ ∫ 1 0 ( Φ(α) + qΦ(Υ) [2]q − φqΘ(Υ− α)2 [2]q (2ϖ − qΘϖ2 −Θϖ2) ) dqϖ = Φ(α) + qΦ(Υ) [2]q − φqΘ(Υ− α)2 [2]q ( 2 [2]q − qΘ [3]q − Θ [3]q ) . The inequality of (18) can be computed as Φ ( α+ qΥ [2]q ) ≤ ∫ 1 0 ( Φ(qΘϖΥ+ (1− qΘϖ)α) [2]q + qΦ(Θϖα+ (1−Θϖ)Υ) [2]q ) dqϖ = (1− q) ∞∑ n=0 qn Φ(Θqn+1Υ+ (1− qn+1Θ)α) + qΦ(qnΘα+ (1−Θqn)Υ) [2]q = 1 Θ[2]q(Υ− α) ( 1 q ∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + q ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − (1− q) q[2]q Φ(ΘΥ+ (1−Θ)α). So, we can write Φ ( α+ qΥ [2]q ) ≤ 1 Θ[2]q(Υ− α) ( 1 q ∫ ΘΥ+(1−Θ)a α Φ(ϖ)αdqϖ C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 10 of 24 +q ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − (1− q) q[2]q f(ΘΥ + (1−Θ)α) ≤ Φ(α) + qΦ(Υ) [2]q − φqΘ(Υ− α)2 [2]q ( 2 [2]q − qΘ [3]q − Θ [3]q ) . (19) Alternatively, by employing a comparable technique, we can formulate Φ ( qα+Υ [2]q ) ≤ ( qΦ(ΘϖΥ+ (1−Θϖ)α) + Φ(qΘϖα+ (1− qΘϖ)Υ) [2]q ) ≤ qΦ(α) + Φ(Υ) [2]q − φqΘ(Υ− α)2 [2]q (2ϖ − qΘϖ2 −Θϖ2). The process of q-integrating both sides of the inequality above yields Φ ( qα+Υ [2]q ) ≤ 1 Θ[2]q(Υ− α) ( q ∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + 1 q ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − (1− q) q[2]q Φ(Θα+ (1−Θ)Υ) ≤ qΦ(α) + Φ(Υ) [2]q − φqΘ(Υ− α)2 [2]q ( 2 [2]q − qΘ [3]q − Θ [3]q ) . (20) Note that 2/[2]q − qΘ/[3]q − Θ/[3]q ≥ 0. By combinating the inequalities (19) and (20), we obtain Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q ) ≤ 1 Θ[2]q(Υ− α) ( ( 1 q + q) ∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ + ( 1 q + q) ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) − (1− q) q[2]q (Φ(ΘΥ + (1−Θ)α) + Φ(Θα+ (1−Θ)Υ)) ≤ Φ(α) + Φ(Υ)− 2φqΘ(Υ− α)2 [2]q ( 2 [2]q − qΘ [3]q − Θ [3]q ) ≤ Φ(α) + Φ(Υ). Multiplying the inequality above by 1/2 results in the third, fourth, and fifth inequalities of (17). The following result gives the refinements of inequalities (5). Corollary 1. Setting Θ = 1/2 in Theorem 5, then we obtain Φ ( α+Υ 2 ) ≤ Φ1(q) ≤ Φ2(q) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 11 of 24 ≤ 1 2(Υ− α) (∫ α+Υ 2 α Φ(ϖ)αdqϖ + ∫ Υ α+Υ 2 Φ(ϖ) Υdqϖ ) ≤ Φ3(q) ≤ Φ(α) + Φ(Υ) 2 , (21) where Φ1(q) = q[2]q (1 + q2) ( Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ) + (1− q) (1 + q2) Φ ( α+Υ 2 ) , Φ2(q) = q[2]q 2(1 + q2) ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) + (1− q) (1 + q2) Φ ( α+Υ 2 ) and Φ3(q) = q[2]q (1 + q2) ( Φ(α) + Φ(Υ) 2 − φq(Υ− α)2 4([2]q)2[3]q ( 3 + 2q + 3q2 )) + (1− q) (1 + q2) Φ ( α+Υ 2 ) . Proof. Setting Θ = 1/2 in Theorem 5, gives us Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) ≤ 1 + q2 q[2]q(Υ− α) (∫ α+Υ 2 α Φ(ϖ)αdqϖ + ∫ Υ α+Υ 2 Φ(ϖ) Υdqϖ ) − 1− q q[2]q Φ ( α+Υ 2 ) ≤ Φ(α) + Φ(Υ) 2 − φq(Υ− a)2 4([2]q)2[3]q ( 3 + 2q + 3q2 ) ≤ Φ(α) + Φ(Υ) 2 . This implies that( 1 + q2 q[2]q ) Φ ( α+Υ 2 ) = Φ ( α+Υ 2 ) + (1− q) q[2]q Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + (1− q) q[2]q Φ ( a+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) + (1− q) q[2]q Φ ( α+Υ 2 ) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 12 of 24 ≤ 1 + q2 q[2]q(Υ− α) (∫ α+Υ 2 α Φ(ϖ)αdqϖ + ∫ Υ α+Υ 2 Φ(ϖ) Υdqϖ ) − (1− q) q[2]q Φ ( α+Υ 2 ) + (1− q) q[2]q Φ ( α+Υ 2 ) ≤ Φ(α) + Φ(Υ) 2 − φq(Υ− α)2 4([2]q)2[3]q ( 3 + 2q + 3q2 ) + (1− q) q[2]q Φ ( α+ b 2 ) ≤ Φ(α) + Φ(Υ) 2 + 1− q q[2]q ( Φ(α) + Φ(Υ) 2 ) = 1 + q2 q[2]q ( Φ(α) + Φ(b) 2 ) . (22) Multiply the inequality (22) by q[2]q/(1 + q2) leads us to the desired result. The result presented below provides refinements for the inequalities (3). Corollary 2. Setting Θ = 1 in Theorem 5, then we obtain Φ ( α+Υ 2 ) ≤ Φ4(q) ≤ Φ5(q) ≤ 1 2(Υ− α) (∫ Υ α Φ(ϖ)αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) ≤ Φ6(q) ≤ Φ(α) + Φ(Υ) 2 , (23) where Φ4(q) = q[2]q (1 + q2) ( Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ) + (1− q) (1 + q2) Φ ( α+Υ 2 ) , Φ5(q) = q[2]q 2(1 + q2) ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) + (1− q) (1 + q2) Φ ( α+Υ 2 ) and Φ6(q) = q[2]q (1 + q2) ( Φ(α) + Φ(Υ) 2 ) − φq2(Υ− α)2 [2]q[3]q + (1− q) (1 + q2) ( Φ(α) + Φ(Υ) 2 ) . Proof. Setting Θ = 1 in Theorem 5 gives us Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 13 of 24 ≤ 1 + q2 2q[2]q(Υ− α) (∫ Υ α Φ(ϖ)αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) − 1− q 2q[2]q (Φ(α) + Φ(Υ)) ≤ Φ(α) + Φ(Υ) 2 − φq(1 + q2)(Υ− α)2 ([2]q)2[3]q ≤ Φ(α) + Φ(Υ) 2 . Which gives( 1 + q2 q[2]q ) Φ ( α+Υ 2 ) = Φ ( α+Υ 2 ) + (1− q) q[2]q Φ ( α+Υ 2 ) ≤ Φ ( α+Υ 2 ) + φ 4 ( (1− q)(Υ− α) [2]q )2 + (1− q) q[2]q Φ ( α+Υ 2 ) ≤ 1 2 ( Φ ( α+ qΥ [2]q ) +Φ ( qα+Υ [2]q )) + (1− q) q[2]q Φ ( α+Υ 2 ) ≤ 1 + q2 2q[2]q(Υ− α) (∫ Υ α f(ϖ)αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) − (1− q) q[2]q ( Φ(α) + Φ(Υ) 2 ) + (1− q) q[2]q ( Φ(α) + Φ(Υ) 2 ) ≤ Φ(α) + Φ(Υ) 2 − φq(1 + q2)(Υ− α)2 ([2]q)2[3]q + (1− q) q[2]q ( Φ(α) + Φ(Υ) 2 ) ≤ Φ(α) + Φ(Υ) 2 + (1− q) q[2]q ( Φ(α) + Φ(Υ) 2 ) = ( 1 + q2 q[2]q )( Φ(α) + Φ(Υ) 2 ) . (24) Multiplying the inequality (24) by q[2]q/(1 + q2) yields the desired result. 4. Parameterized q-integral inequalities We prove the H-H inequalities (11) of Theorem 4 by utilizing the q-differentiability of the function. Lemma 1 ([2]). Let Φ : [α,Υ] → R be a q-differentiable function. If αDqΦ and ΥDqΦ are two continuous and integrable functions on [α,Υ], then we have: Θ(Υ− α) 2 ∫ 1 0 qϖ ( ΥDqΦ(Θϖα+ (1−Θϖ)Υ)− αDqΦ(ΘϖΥ+ (1−Θϖ)α) ) dqϖ C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 14 of 24 = 1 2Θ(Υ− α) (∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ + ∫ ΘΥ+(1−Θ)α α Φ(ϖ)αdqϖ ) − Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 . (25) Proof. By Definition 3, we have I1 := ∫ 1 0 qϖ ( ΥDqΦ(Θϖα+ (1−Θϖ)Υ) ) dqϖ = ∫ 1 0 qϖ Φ(qΘϖα+ (1− qΘϖ)Υ)− Φ(Θϖα+ (1−Θϖ)Υ) (1− q)Θϖ(Υ− α) dqϖ = q (Υ− α)Θ ∞∑ n=0 qnΦ(Θqn+1α+ (1−Θqn+1)Υ) − q (Υ− α)Θ ∞∑ n=0 qnΦ(Θqnα+ (1−Θqn)Υ) = 1 (Υ− α)Θ ∞∑ n=0 qnΦ(Θqnα+ (1−Θqn)Υ)− 1 (Υ− α)Θ Φ(Θα+ (1−Θ)Υ) − q (Υ− α)Θ ∞∑ n=0 qnΦ(Θqnα+ (1−Θqn)Υ) = 1− q (Υ− α)Θ ∞∑ n=0 qnΦ(Θqnα+ (1−Θqn)Υ)− 1 (Υ− α)Θ Φ(Θα+ (1−Θ)b) = 1 (Υ− α)2Θ2 ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ − 1 (Υ− α)Θ Φ(Θα+ (1−Θ)Υ). Similarly, by Definition 2, we have I2 := ∫ 1 0 qϖ ( αDqΦ(ΘϖΥ+ (1−Θϖ)α)) dqϖ = 1 (Υ− α)Θ Φ(ΘΥ+ (1−Θ)α)− 1 (Υ− α)2Θ2 ∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ. Then it follows that Θ(Υ− α) 2 (I1 − I2) = Θ(Υ− α) 2 ( 1 (Υ− α)2Θ2 ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ − 1 (Υ− α)Θ Φ(Θα+ (1−Θ)Υ) − 1 (Υ− α)Θ Φ(ΘΥ+ (1−Θ)α) + 1 (Υ− α)2Θ2 ∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ ) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 15 of 24 = Θ(Υ− α) 2 (∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ + ∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ ) − Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 . This finalizes the proof. Theorem 6. Let Φ : [α,Υ] → R be a q-differentiable function. If |αDqΦ| and | ΥDqΦ| are strongly convex functions on [α,Υ] for φ > 0. Then the following inequalities are established:∣∣∣∣∣Θ(Υ− α) 2 (∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) −Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 ∣∣∣∣ ≤ Θq(Υ− α) 2[2]q[3]q ( [2]qΘ ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + ([3]q − [2]qΘ) ( | ΥDqΦ(Υ)| +| αDqΦ(α)|) ) −Θ2qφ(Υ− α)3 ( [4]q − [3]qΘ [3]q[4]q ) ≤ Θq(Υ− α) 2[2]q[3]q ( [2]qΘ ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + ([3]q − [2]qΘ) ( | ΥDqΦ(Υ)| +| αDqΦ(α)|) ) . (26) Proof. It follows from Lemma 1 and |αDqΦ| and |ΥDqΦ| are strongly convex functions that ∣∣∣∣∣Θ(Υ− α) 2 (∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ + ∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ ) −Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 ∣∣∣∣ ≤ Θ(Υ− α) 2 ∫ 1 0 qϖ ∣∣ ΥDqΦ(Θϖα+ (1−Θϖ)Υ) ∣∣ dqϖ + Θ(Υ− α) 2 ∫ 1 0 qϖ | αDqΦ(ΘϖΥ+ (1−Θϖ)α)| dqϖ ≤ Θ(Υ− α) 2 ∫ 1 0 qϖ ( Θϖ| ΥDqΦ(α)|+ (1−Θϖ)| ΥDqf(Υ)| − φΘϖ(1−Θϖ)(Υ− α)2 ) dqϖ + Θ(Υ− α) 2 ∫ 1 0 qϖ ( Θϖ| αDqΦ(Υ)| C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 16 of 24 + (1−Θϖ)| αDqΦ(α)| − φΘϖ(1−Θϖ)(Υ− α)2 ) dqϖ = Θq(Υ− α) 2[2]q[3]q ( [2]qΘ ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + ([3]q − [2]qΘ) ( | ΥDqΦ(Υ)| +| αDqΦ(α)|) ) −Θ2qφ(Υ− α)3 ( [4]q − [3]qΘ [3]q[4]q ) ≤ Θq(Υ− α) 2[2]q[3]q ( [2]qΘ ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + ([3]q − [2]qΘ) ( | ΥDqΦ(Υ)| +| αDqΦ(α)|) ) . This finalizes the proof. Remark 3. When Θ = 1 in Theorem 6, we derive trapezoid-type inequalities:∣∣∣∣∣(Υ− α) 2 (∫ Υ α Φ(ϖ) αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) − Φ(α) + Φ(Υ) 2 ∣∣∣∣∣ ≤ q(Υ− α) 2[2]q[3]q ( [2]q ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + q2 ( | ΥDqΦ(Υ)|+ | αDqΦ(α)| )) − q4φ(Υ− α)3 [3]q[4]q ≤ q(Υ− α) 2[2]q[3]q ( [2]q ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + q2 ( | ΥDqΦ(Υ)|+ | αDqΦ(α)| )) . Remark 4. When Θ = 1/2 in Theorem 6, we derive Midpoint-type inequalities:∣∣∣∣∣(Υ− α) 2 (∫ (α+Υ) 2 α Φ(ϖ) αdqϖ + ∫ Υ (α+Υ) 2 Φ(ϖ) Υdqϖ ) − Φ ( α+Υ 2 )∣∣∣∣∣ ≤ q(Υ− α) 8[2]q[3]q ( [2]q ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + ([3]q + q2) ( | ΥDqΦ(Υ)| +| αDqΦ(α)|) ) − q4φ(Υ− α)3 ( q3 + [4]q [3]q[4]q ) ≤ q(Υ− α) 8[2]q[3]q ( [2]q ( | ΥDqΦ(α)|+ | αDqΦ(Υ)| ) + ([3]q + q2) ( | ΥDqΦ(Υ)| +| αDqΦ(α)|) ) . C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 17 of 24 Theorem 7. Let Φ : [α,Υ] → R be a q-differentiable function. If |αDqΦ|c and | ΥDqΦ|c, c > 1 are strongly convex functions on [α,Υ] for φ > 0 , then the following inequalities are established:∣∣∣∣∣Θ(Υ− α) 2 (∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) −Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 ∣∣∣∣ ≤ qΘ(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( Θ| αDqΦ(Υ)|c [2]q + ( 1− Θ [2]q ) | αDqΦ(α)|c − φΘ ( 1 [2]q − Θ [3]q ) (Υ− α)2 ) 1 c + ( Θ| ΥDqΦ(α)|c [2]q + ( 1− Θ [2]q ) | ΥDqΦ(Υ)|c − φΘ ( 1 [2]q − Θ [3]q ) (Υ− α)2 ) 1 c ) ≤ qΘ(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( Θ| αDqf(Υ)|c [2]q + ( [2]q −Θ [2]q ) | αDqf(α)|c ) 1 c + ( Θ| ΥDqΦ(α)|c [2]q + ( [2]q −Θ [2]q ) | ΥDqΦ(Υ)|c ) 1 c ) , (27) where 1/c+ 1/d = 1. Proof. It follows from Lemma 1, Hölder’s inequality, and |αDqΦ|c and |ΥDqΦ|c are strongly convex functions that∣∣∣∣∣Θ(Υ− α) 2 (∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) −Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 ∣∣∣∣ ≤ Θ(Υ− α) 2 (∫ 1 0 (qϖ)ddqϖ ) 1 d (∫ 1 0 | αDqΦ(Θϖb+ (1−Θϖ)α)|c dqϖ ) 1 c + Θ(Υ− α) 2 (∫ 1 0 (qϖ)ddqϖ ) 1 d (∫ 1 0 ∣∣ ΥDqΦ(Θϖα+ (1−Θϖ)Υ) ∣∣c dqϖ) 1 c ≤ Θ(Υ− α) 2 (∫ 1 0 (qϖ)ddqϖ ) 1 d (∫ 1 0 (Θϖ| αDqΦ(Υ)|c + (1−Θϖ)| αDqΦ(α)|c C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 18 of 24 − φΘϖ(1−Θϖ)(Υ− α)2)dqϖ ) 1 c + Θ(Υ− α) 2 (∫ 1 0 (qϖ)ddqϖ ) 1 d (∫ 1 0 (Θϖ | bDqΦ(α)|c + (1−Θϖ)| ΥDqΦ(Υ)|c − φΘϖ(1−Θϖ)(b− α)2)dqϖ ) 1 c ≤ qΘ(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( Θ| αDqΦ(Υ)|c [2]q + ( 1− Θ [2]q ) | αDqΦ(α)|c ) 1 c + ( Θ| ΥDqΦ(α)|c [2]q + ( 1− Θ [2]q ) | ΥDqΦ(Υ)|c ) 1 c ) ≤ qΘ(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( Θ| αDqΦ(Υ)|c [2]q + ( [2]q −Θ [2]q ) | αDqΦ(α)|c ) 1 c + ( Θ| ΥDqΦ(α)|c [2]q + ( [2]q −Θ [2]q ) | ΥDqΦ(Υ)|c ) 1 c ) . This concludes the proof. Remark 5. When Θ = 1 in Theorem 7, we derive trapezoid-type inequalities:∣∣∣∣∣(Υ− α) 2 (∫ Υ α Φ(ϖ) αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) − Φ(α) + Φ(Υ) 2 ∣∣∣∣∣ ≤ q(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( | αDqΦ(Υ)|c [2]q + ( 1− 1 [2]q ) | αDqΦ(α)|c − φ ( 1 [2]q − 1 [3]q ) (Υ− α)2 ) 1 c + ( | ΥDqΦ(α)|c [2]q + ( 1− 1 [2]q ) | ΥDqΦ(Υ)|c − φ ( 1 [2]q − 1 [3]q ) (Υ− α)2 ) 1 c ) ≤ q(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( | αDqΦ(Υ)|c + q| αDqΦ(α)|c [2]q ) 1 c + ( | ΥDqΦ(α)|c + q| ΥDqΦ(Υ)|c [2]2 ) 1 c ) . Remark 6. When Θ = 1/2 in Theorem 7, we derive Midpoint-type inequalities:∣∣∣∣∣(Υ− α) 2 (∫ (α+Υ) 2 α Φ(ϖ) αdqϖ + ∫ Υ (α+Υ) 2 Φ(ϖ) Υdqϖ ) − Φ ( α+Υ 2 )∣∣∣∣∣ C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 19 of 24 ≤ q(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( | αDqΦ(Υ)|c 2[2]q + ( 1− 1 2[2]q ) | αDqΦ(α)|c − φ 2 ( 1 [2]q − 1 2[3]q ) (Υ− α)2 ) 1 c + ( | ΥDqΦ(α)|c 2[2]q + ( 1− 1 2[2]q ) | ΥDqΦ(Υ)|c − φ 2 ( 1 [2]q − 1 2[3]q ) (Υ− α)2 ) 1 c ) ≤ q(Υ− α) 4 ( 1 [d+ 1]q ) 1 d (( | αDqΦ(Υ)|c + ([2]q + q)| αDqΦ(α)|c [2]q ) 1 c + ( | ΥDqΦ(α)|c + ([2]q + q)| ΥDqΦ(Υ)|c [2]2 ) 1 c ) . Theorem 8. Let Φ : [α,Υ] → R be a q-differentiable function. If |αDqΦ|c and | ΥDqΦ|c, c ≥ 1 are strongly convex functions on [α,Υ] for φ > 0 , then the following inequalities are established:∣∣∣∣∣Θ(Υ− α) 2 (∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) −Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 ∣∣∣∣ ≤ qΘ(Υ− α) 2 ( 1 [2]q )1− 1 c (( Θ| αDqf(Υ)|c [3]q + ( 1 [2]q − Θ [3]q ) | αDqΦ(α)|c − φΘ ( 1 [3]q − Θ [4]q ) (Υ− α)2 ) 1 c + ( Θ| ΥDqΦ(α)|c [3]q + ( 1 [2]q − Θ [3]q ) | ΥDqΦ(Υ)|c − φΘ ( 1 [3]q − Θ [4]q ) (Υ− α)2 ) 1 c ) ≤ qΘ(Υ− α) 2[2]q (( [2]qΘ| αDqΦ(Υ)|c + ([3]q −Θ[2]q)| αDqΦ(α)|c [3]q ) 1 c + ( [2]qΘ| ΥDqΦ(α)|c + ([3]q −Θ[2]q)| ΥDqΦ(Υ)|c [3]q ) 1 c ) . (28) Proof. It can be deduced from Lemma 1, the power mean inequality and and the strong convexity of |αDqΦ|c and |ΥDqΦ|c that∣∣∣∣∣Θ(Υ− α) 2 (∫ ΘΥ+(1−Θ)α α Φ(ϖ) αdqϖ + ∫ Υ Θα+(1−Θ)Υ Φ(ϖ) Υdqϖ ) C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 20 of 24 −Φ(Θα+ (1−Θ)Υ) + Φ(ΘΥ+ (1−Θ)α) 2 ∣∣∣∣ ≤ Θ(Υ− α) 2 (∫ 1 0 qϖ dqϖ )1− 1 c (∫ 1 0 qϖ | αDqΦ(ΘϖΥ+ (1−Θϖ)α)|c dqϖ ) 1 c + Θ(Υ− α) 2 (∫ 1 0 qϖ dqϖ )1− 1 c (∫ 1 0 qϖ ∣∣ ΥDqΦ(Θϖα+ (1−Θϖ)Υ) ∣∣c dqϖ) 1 c ≤ Θ(Υ− α) 2 (∫ 1 0 qϖ dqϖ )1− 1 c (∫ 1 0 qϖ(Θϖ| αDqΦ(Υ)|c + (1−Θϖ)| αDqΦ(α)|c − φΘϖ(1−Θϖ)(Υ− α)2)dqϖ ) 1 c + Θ(Υ− α) 2 (∫ 1 0 qϖ dqϖ )1− 1 c (∫ 1 0 qϖ (Θϖ| ΥDqΦ(α)|c + (1−Θϖ)| ΥDqΦ(Υ)|c − φΘϖ(1−Θϖ)(Υ− α)2)dqϖ ) 1 c = Θ(Υ− α) 2 ( q [2]q )1− 1 c ( qΘ| αDqΦ(Υ)|c [3]q + ( q [2]q − qΘ [3]q ) | αDqΦ(α)|c − φqΘ ( 1 [3]q − Θ [4]q ) (Υ− α)2 ) 1 c + Θ(Υ− α) 2 ( q [2]q )1− 1 c ( qΘ| ΥDqΦ(α)|c [3]q + ( q [2]q qΘ [3]q ) | ΥDqΦ(Υ)|c − φqΘ ( 1 [3]q − Θ [4]q ) (Υ− α)2 ) 1 c ≤ qΘ(Υ− α) 2[2]q (( [2]qΘ| αDqΦ(Υ)|c + ([3]q −Θ[2]q)| αDqΦ(α)|c [3]q ) 1 c + ( [2]qΘ| ΥDqΦ(α)|c + ([3]q −Θ[2]q)| ΥDqΦ(Υ)|c [3]q ) 1 c ) . This concludes the proof. Remark 7. When Θ = 1 in Theorem 8, we derive trapezoid-type inequalities:∣∣∣∣∣(Υ− α) 2 (∫ Υ α Φ(ϖ) αdqϖ + ∫ Υ α Φ(ϖ) Υdqϖ ) − Φ(α) + Φ(Υ) 2 ∣∣∣∣∣ ≤ q(Υ− α) 2 ( 1 [2]q )1− 1 c (( | αDqΦ(Υ)|c [3]q + ( 1 [2]q − 1 [3]q ) | αDqΦ(α)|c C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 21 of 24 − φ ( 1 [3]q − 1 [4]q ) (Υ− α)2 ) 1 c + ( | ΥDqΦ(α)|c [3]q + ( 1 [2]q − 1 [3]q ) | ΥDqΦ(Υ)|c − φ ( 1 [3]q − 1 [4]q ) (Υ− α)2 ) 1 c ) ≤ q(Υ− α) 2[2]q (( [2]qΘ| αDqΦ(Υ)|c + q2| αDqΦ(α)|c [3]q ) 1 c + ( [2]q| ΥDqΦ(α)|c + q2| ΥDqΦ(Υ)|c [3]q ) 1 c ) . Remark 8. When Θ = 1/2 in Theorem 8, we derive Midpoint-type inequalities:∣∣∣∣∣(Υ− α) 2 (∫ (α+Υ) 2 α Φ(ϖ) αdqϖ + ∫ Υ (α+Υ) 2 Φ(ϖ) Υdqϖ ) − Φ ( α+Υ 2 )∣∣∣∣∣ ≤ q(Υ− α) 2 ( 1 [2]q )1− 1 c (( | αDqΦ(b)|c 2[3]q + ( 1 [2]q − 1 2[3]q ) | αDqΦ(α)|c − φ 2 ( 1 [3]q − 1 2[4]q ) (Υ− α)2 ) 1 c + ( | ΥDqΦ(α)|c 2[3]q + ( 1 [2]q − 1 2[3]q ) | ΥDqΦ(Υ)|c − φ 2 ( 1 [3]q − 1 2[4]q ) (Υ− α)2 ) 1 c ) ≤ q(Υ− α) 2[2]q (( [2]q| αDqΦ(Υ)|c + ([3]q + q2)| αDqΦ(α)|c 2[3]q ) 1 c + (( [2]q| ΥDqΦ(α)|c + ([3]q + q2)| ΥDqΦ(Υ)|c 2[3]q ) 1 c . 5. Applications We show how special means can be used to prove the results in Theorems 6, 7, and 8. To establish arbitrary positive numbers W1 and W2 (W1 ̸= W2), we define the means as follows: (i) The arithmetic mean A = A(W1,W2) = W1 +W2 2 . C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 22 of 24 (ii) The logarithmic mean LP = LP (W1,W2) = WP+1 2 −WP+1 1 (P + 1)(W1 −W2) . Proposition 1. Given that 0 < α < Υ, the following inequalities are valid:∣∣∣∣ 1 c+ 1 ( Θ2(Υ− α)2A(k1, k2)−A(W c+1 1 ,W c+1 2 ) )∣∣∣∣ ≤ qΘ(Υ− α) 2[2]q[3]q ( [2]qΘ(Lc(qα+ (1− q)Υ, α) + Lc(qΥ+ (1− q)α,Υ)) + ([3]q − [2]qΘ)(αc +Υc) ) − Θ2qφ(Υ− α)3([4]q − [3]qΘ) [3]q[4]q , where k1 = (1− q) ∞∑ n=0 qn (qnΘΥ+ (1− qnΘ)α)c+1 k2 = (1− q) ∞∑ n=0 qn (qnΘα+ (1− qnΘ)Υ)c+1 and W1 = Θα+ (1−Θ)Υ, W2 = ΘΥ+ (1−Θ)α. Proof. By substituting Φ(ϖ) = ϖc+1/(c + 1), where ϖ > 0 in the inequalities (26) of Theorem 6, then we obtain the result. Proposition 2. Given that 0 < α < Υ, the following inequalities are valid:∣∣∣∣ 1 c+ 1 ( Θ2(Υ− α)2A(k1, k2)−A(W c+1 1 ,W c+1 2 ) )∣∣∣∣ ≤ qΘ(Υ− α) 2 ( 1 [d+ 1]q ) 1 d (( Θ|Lc(qΥ+ (1− q)α,Υ)|c [2]q + ( 1− Θ [2]q ) αc − φΘ( 1 [2]q − Θ [3]q )(Υ− α)2 ) 1 c + ( Θ|Lc(qα+ (1− q)Υ, α)|c [2]q + ( 1− Θ [2]q ) Υc − φΘ ( 1 [2]q − Θ [3]q ) (Υ− α)2 ) 1 c ) . Proof. By substituting Φ(ϖ) = ϖc+1/(c + 1), where ϖ > 0 in the inequalities (27) of Theorem 7, we obtain the result. C. Sahatsathatsana, P. Yotkaew / Eur. J. Pure Appl. Math, 18 (1) (2025), 5661 23 of 24 Proposition 3. Given that 0 < α < Υ, the following inequalities are valid:∣∣∣∣ 1 c+ 1 ( Θ2(Υ− α)2A(k1, k2)−A(W c+1 1 ,W c+1 2 ) )∣∣∣∣ ≤ qΘ(Υ− α) 2 ( 1 [2]q )1− 1 c (( Θ|Lc(qΥ+ (1− q)α,Υ)|c [3]q + ( 1 [2]q − Θ [3]q ) αc − φΘ ( 1 [3]q − Θ [4]q ) (b− α)2 ) 1 c + ( Θ|Lc(qα+ (1− q)Υ, α)|c [2]q + ( 1 [2]q − Θ [3]q ) Υc − φΘ ( 1 [3]q − Θ [4]q ) (Υ− α)2 ) 1 c ) . Proof. By substituting Φ(ϖ) = ϖc+1/(c + 1), where ϖ > 0 in the inequalities (28) of Theorem 8, we obtain the result. 6. Conclusion This study focused on using the principles of q-calculus to prove varieties of q-H-H type inequalities for strongly convex functions. Our findings indicate the potential for generalizing existing comparable results in the literature. In addition, we also refine the results in the literature of N. Alp et al. [2] by using the characterization of strong convexity of Φ. 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